Spontaneous Symmetry Breaking in Field Theory

Spontaneous symmetry breaking occurs when a system's lowest-energy state fails to respect a symmetry of its underlying laws, a mechanism central to particle physics and condensed matter alike.

Definition

Spontaneous symmetry breaking is the situation in which the equations or Lagrangian of a theory possess a symmetry that the actual ground state does not share, so that the symmetry is hidden by the choice of vacuum rather than absent from the dynamics.

Scope

This topic covers the general phenomenon in which the Lagrangian of a field theory is symmetric but its ground state is not, leading to a degenerate set of vacua. It treats Goldstone's theorem, which predicts massless scalar bosons for each spontaneously broken continuous global symmetry, the way these would-be Goldstone bosons are absorbed when the symmetry is gauged, and the broad applications from superconductivity to the electroweak sector.

Core questions

• How can the laws of a theory be symmetric while its ground state is not?
• Why does breaking a continuous global symmetry produce massless Goldstone bosons?
• What happens to the Goldstone bosons when the broken symmetry is a gauge symmetry?
• How does the same mechanism appear in superconductivity and in particle physics?

Key concepts

• Degenerate vacua
• Order parameter and vacuum expectation value
• Goldstone bosons
• Chiral symmetry breaking
• Hidden symmetry
• Connection to the Higgs mechanism

Key theories

Goldstone theorem

Goldstone showed that the spontaneous breaking of a continuous global symmetry produces one massless scalar boson for each broken generator, a result that constrains the spectrum of broken-symmetry theories.

Dynamical symmetry breaking

Nambu and Jona-Lasinio demonstrated, by analogy with superconductivity, that interactions can dynamically generate fermion masses and break chiral symmetry spontaneously, with associated near-massless bosons such as the pion.

Mechanisms

When a field potential has a continuous set of minima rather than a single symmetric one, the system must select one minimum, and small excitations along the flat directions of the potential cost no energy, appearing as massless Goldstone bosons. If the broken symmetry is local rather than global, these massless modes are not physical but instead become the longitudinal components of the gauge bosons, which thereby acquire mass through the Higgs mechanism.

Clinical relevance

Spontaneous symmetry breaking underlies the Higgs mechanism that gives mass to the electroweak gauge bosons, explains the lightness of the pion as an approximate Goldstone boson of chiral symmetry breaking, and provides a unifying concept linking particle physics to superconductivity, magnetism, and other phase transitions.

History

The idea that a symmetric theory could have an asymmetric ground state was imported into particle physics from the theory of superconductivity around 1960 by Nambu, who applied it to dynamical mass generation. Goldstone's theorem of 1961 established the appearance of massless bosons, and the resolution of how to evade them in gauge theories led directly to the Higgs mechanism and the electroweak theory, with Nambu honored by the 2008 Nobel Prize.

Key figures

• Jeffrey Goldstone
• Yoichiro Nambu
• Philip Anderson
• Steven Weinberg

Seminal works

• goldstone1961
• nambu1961

Frequently asked questions

What is a Goldstone boson?

A Goldstone boson is a massless scalar particle that appears when a continuous global symmetry is spontaneously broken, with one such boson for each broken symmetry generator. In gauge theories these modes are absorbed by the gauge bosons.

Is spontaneous symmetry breaking unique to particle physics?

No. It is a general phenomenon that also describes ferromagnetism, where rotational symmetry is broken by a magnetization direction, and superconductivity, from which the particle-physics applications were originally borrowed.